Symmetry and Stability in Taylor-Couette Flow · Introduction. The flow of a fluid between...

SIAM J. MATH. ANAL.Vol. 17, No. 2, March 1986

(C) 1986 Society for Industrial and Applied Mathematics001

SYMMETRY AND STABILITY INTAYLOR-COUETTE FLOW*

MARTIN GOLUBITSKY" AND IAN STEWART $

Abstract. We study the flow of a fluid between concentric rotating cylinders (the Taylor problem) byexploiting the symmetries of the system. The Navier-Stokes equations, linearized about Couette flow, possesstwo zero and four purely imaginary eigenvalues at a suitable value of the speed of rotation of the outercylinder. There is thus a reduced bifurcation equation on a six-dimensional space which can be shown tocommute with an action of the symmetry group 0(2)S0(2). We use the group structure to analyze thisbifurcation equation in the simplest (nondegenerate) case and to compute the stabilities of solutions. Inparticular, when the outer cylinder is counterrotated we can obtain transitions which seem to agree withrecent experiments of Andereck, Liu, and Swinney [1984]. It is also possible to obtain the "main sequence" inthis model. This sequence is normally observed in experiments when the outer cylinder is held fixed.

Introduction. The flow of a fluid between concentric rotating cylinders, orTaylor-Couette flow, is known to exhibit a variety of types of behavior, the mostcelebrated being Taylor vortices (Taylor [1923]). The problem has been studied by alarge number of authors: a recent survey is that of DiPrima and Swinney [1981]. Theexperimental apparatus has circular symmetry, and the standard mathematical idealiza-tion (periodic boundary conditions at the ends of the cylinder) introduces a furthersymmetry. As a result the Navier-Stokes equations for this problem are covariant withrespect to the action of a symmetry group 0(2)SO(2). It has become clear that thesymmetries inherent in bifurcating systems have a strong influence on their behavior. Inthis paper we study a series of bifurcations that occur in Taylor-Couette flow placingemphasis on the role of symmetry. (Schecter [1976] and Chossat and Iooss [1984] havealso studied the problem from this viewpoint, and we discuss the relations between ourwork and theirs below.)

DiPrima and Grannick [1971] have found that when the outer cylinder is rotatedin a direction opposite to that of the inner cylinder, the Navier-Stokes equations,linearized about Couette flow, possess six eigenvalues on the imaginary axis. It followsthat aspects of the dynamics can be reduced (either by Lyapunov-Schrnidt or centermanifold reduction) to a vector field on R6; furthermore, this vector field commuteswith an action of 0(2)S0(2). Moreover, as we explain in 7, recent experimentalresults due to Andereck, Liu, and Swinney [1984] seem to confirm the existence of thesix-dimensional kernel.

We point out in particular that the six-dimensional kernel is a codimension onephenomenon, and hence it is not surprising that it should be possible to find it byvarying only one parameter. Indeed, this degeneracy should occur relatively often invarious circumstances, and so deserves detailed analysis.

We study the general class of bifurcation problems on R6 having this 0(2) SO(2)symmetry. We derive the general form possible for the vector field, and by classifying

Received by the editors November 15, 1984, and in revised form February 8, 1985.Department. of Mathematics, University of Houston-University Park, Houston, Texas 77004. The

research of this author was supported in part by the National Science Foundation under grant MCS-8101580,and by grant NAG 2-279 from NASA-Ames.

*Department of Mathematics, University of Houston-University Park, Houston, Texas 77004 andMathematics Institute, University of Warwick, Coventry, CV4 7AL, England. The research of this author wassupported in part by grant NAG 2-279 from NASA-Ames.

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250 MARTIN GOLUBITSKY AND IAN STEWART

the possible ways to break symmetry, obtain equations for the bifurcating branches(subject to certain nondegeneracy conditions). We also obtain the (linearized orbital)stabilities of these branches.

By introducing an additional parameter et we split the kernel R6 into two subspacesR2 and R4 corresponding to a steady-state and a periodic bifurcation respectively.Depending on the sign of a, one or other of these bifurcations occurs first.

By inspecting the symmetries of the physically observed solutions we may tenta-tively identify them with various branches: in particular the flows known as Taylorvortices, wavy vortices, twisted vortices, helices (or spirals) seem to correspond natu-rally to solution branches; and there is also a branch described by DiPrima andGrannick [1971] as the "nonaxisymmetric simple mode".

The experimental results of Andereck, Liu, and Swinney may be summarized asfollows. In the weakly counterrotating case (that is, when the speed of the outercylinder 2o is slightly less than the critical speed 2 where the six-dimensional kernelappears) the following transition sequence is observed as 2i, the speed of the innercylinder, is increased.

Couette flow Taylor vortices wavy vortices

where the final state obtained when the wavy vortices lose stability seems not to be onerepresentable in the six-dimensional kernel. In the strongly counterrotating case (thatis, when f0 is slightly greater than fl) the observed transition sequence is:

Couette flow ---, spiral cells wavy spiral cells.

We shall show in [}7 that it is possible to make a nondegenerate choice of vectorfields on R6 having 0(2)S0(2) symmetry which produces the same transition se-quences in the following sense. It is possible to determine constraints on the Taylorexpansion of this vector field, given only by inequalities on coefficients in this Taylorexpansion, so that the solutions corresponding to these states are (orbitally) asymptoti-cally stable and lose stability in a way that should produce the desired transitions.Moreover, when these inequalities are satisfied, no other solutions are asymptoticallystable.

We also show in [}7 that it is possible to choose these constraints differently, sothat the "main sequence" of transitions occurs, namely,

Couette flow Taylor vortices wavy vortices

modulated wavy vortices

This transition sequence is usually observed when the outer cylinder is held stationary(20 =0). What we show is that it is possible for the "wavy vortex solutions" to losestability to a torus bifurcation, where two Floquet exponents cross the imaginary axis.This tertiary bifurcation has never been demonstrated theoretically hitherto. At thispoint, however, we cannot prove that the branch of "modulated wavy vortices" isasymptotically stable, though we hope that the results of Scheurle and Marsden [1984]will provide the techniques required to carry out this computation. We do show,moreover, that no other solutions are asymptotically stable when these constraints hold.In particular, stable spiral cells should not occur in this experimental situation.

The paper is organized as follows. In 1 we describe some of the flows observed inthe Taylor experiment and review the evidence for the existence of a six-dimensionalkernel. In [}2 we discuss the symmetries that act on the six-dimensional kernel, and in3 we discuss the symmetries of the observed flows. In [}4 (and the Appendix) we

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SYMMETRY AND STABILITY IN TAYLOR-COUETTE FLOW 251

discuss the reduction procedure and derive the exact form of the reduced mapping (orvector field) on the six-dimensional kernel prescribed by those symmetries. We classifythe (conjugacy classes) of isotropy subgroups (which describe the type of symmetry-breaking that occurs at bifurcations). The heart of the paper is 5, where we analyze thebranching equations and the stability of branches. In 6 we use these to obtain a list ofthe sixteen inequalities that must be imposed to ensure (what we mean by) nondegener-acy. Finally, in 7, we compare the six-dimensional model with experimental observa-tions in both the counterrotating case and the case when the outer cylinder is heldfixed.

1. The Taylor problem. By the term "Taylor problem" we mean the study of boththe possible states of fluid flow between two rotating concentric cylinders, and thetransitions between these states. The Taylor problem provides a beautiful example of abifurcation problem with symmetry. In this paper we discuss how these symmetriesaffect the structure of the bifurcating solutions.

We denote the angular velocities of the inner and outer cylinders by fi and 0respectively. To specify a direction, we assume that i >_-0. In the standard experimentsthe outer cylinder is held fixed (f0 0) and the inner cylinder is speeded up in stagesfrom i =0, at each stage allowing the flow to settle into a stable pattern; see Taylor[1923], Gollub and Swinney [1975]. Experiments have been performed in both thecorotating case (0>0), see Andereck, Dickman and Swinney [1983], and the counter-rotating case (f0 <0), see Andereck, Liu, and Swinney [1984]. (These papers cite theearlier experimental work.) The experiments begin by rotating the outer cylinder atconstant speed, and allowing the flow to stabilize; then the inner cylinder is speeded upas before. The experiments reveal a large number of fluid states, only some of which areunderstood on theoretical grounds. There can exist multiple steady states whose ex-ploration requires different experimental procedures; see for example Coles [1965],Benjamin [1978a, b], Benjamin and Mullin [1982]. In our discussion we shall assume afixed (but unspecified) value of 0, and treat fi (or the corresponding Reynoldsnumber) as a bifurcation parameter. Our main concern will be with the series ofbifurcations that occurs as i is increased steadily. We mention this because manynumerical computations fix the ratio f0/fi, and hence do not correspond directly tothe usual experimental procedure--a fact that, in the presence of multiple states, raisessome problems of interpretation.

In the standard experiments, with f0=0, the first transition is from Couette(laminar) flow to (Taylor) oortices. Both flows are time-independent. This transitionwas first described, in terms of a steady state bifurcation, by Davey [1962]. He showedthat as fi is increased, the Navier-Stokes equations linearized about Couette flow havea double zero eigenvalue at the first bifurcation. At this eigenvalue Couette flow losesstability, and a branch of vortex solutions bifurcates. Davey’s observations have beenreproduced by several authors in different contexts, cf. the survey by DiPrima andSwinney [1981, 6.3]. Note that the appearance of a double zero eigenvalue might besurprising were it not for the existence of symmetries (which can couple eigenvaluestogether and force a degeneracy).

Again, in the standard experiments with 0 0, a second transition is observed, inwhich vortices lose stability to a time-periodic state known as wavy vortices. Presumablythis transition takes place by way of a Hopf type bifurcation in which several eigenval-ues (governing the stability of vortices) cross the imaginary axis as fi is increased.However, this presumption has never been established directly. What has been shown(in Davey, DiPrima, and Stuart [1968]) is that along the Couette branch of solutions

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several eigenvalues of the linearized Navier-Stokes equations cross the imaginary axisas 2 is increased. In particular the next set of eigenvalues to cross the imaginary axis isa complex conjugate pair of purely imaginary eigenvalues, each of multiplicity two.Again, it would be surprising to see four eigenvalues crossing the imaginary axissimultaneously were it not for the symmetry. We note in passing (and amplify theseremarks below) that the 0(2) symmetry which couples these four eigenvalues togetherforces the occurrence of two branches of time-periodic solutions bifurcating from the(unstable) main Couette branch: see Schecter [1976], and Golubitsky and Stewart[1985]. However, neither of these solutions can correspond to wavy vortex states, sincetheir symmetries do not match those of wavy vortices. In fact, one of them has thesymmetries of spiral cells (helices).

There are three additional facts which suggest that there might be a relatit)elysimple local explanation for many of the observed states in the Taylor problem, at leastin the counterrotating case 20 < 0. First, as observed in DiPrima and Grannick [1971],and Krueger, Gross, and DiPrima [1966], there is a critical speed of counterrotation

f < 0 such that, as 2i is increased, Couette flow loses linearized stability by having sixeigenvalues cross the imaginary axis. These six eigenvalues are obtained by amalgamat-ing the double zero eigenvalues and the complex conjugate pair of purely imaginaryeigenvalues of multiplicity two, described above. Further, when 20 is slightly less thanf, the first bifurcation from Couette flow occurs when four eigenvalues (a complexconjugate pair each of multiplicity two) cross the imaginary axis; and there is a doublezero eigenvalue at a higher value of

Second, in experiments in which 20 is sufficiently negative, the primary bifurcationis not to the time-independent Taylor vortices, but to time-dependent spiral cells, seeAndereck, Liu, and Swinney [1984].

Third, it is possible to produce a solution from the interaction of the four-dimensional center manifold (associated with the purely imaginary eigenvalues) and thetwo-dimensional center manifold (associated with the double zero eigenvalues) that hasthe same symmetry as wavy vortices. This suggests that it might be possible to provethe existence of a Hopf-type bifurcation from vortices to wavy vortices as a secondarybifurcation. This was observed by DiPrima and Sijbrand [1982] and again by Chossatand Iooss [1984].

Given these three facts, it would appear reasonable to study the Taylor problem interms of perturbations of the degenerate case f0 2, using either a center manifold ora Lyapunov-Schmidt reduction from the Navier-Stokes equations. We call this degen-eracy the six-dimensional kernel since the linearized equation has a kernel of dimensionsix and the reduced problem may therefore be posed on R6. The hope raised by theabove facts is that one might be able to find a six-dimensional model which explainsthe observed prechaotic states and transitions in the counterrotating Taylor problem.

Let us consider the reduction in more detail. Rigorously, one can use the centermanifold theorem to reduce the (infinite-dimensional) dynamics of the Navier-Stokesequations, near f0 f and near Couette flow, to the study of some vector field g on asix-dimensional center manifold. Alternatively, one can focus only on time-independentand time-periodic solutions and use a reduction of the Lyapunov-Schmidt type toshow the existence of a smooth (i.e. C) mapping h: R6R6 whose zeros are inone-to-one correspondence with the small-amplitude time-periodic (and time-indepen-dent) solutions of the Navier-Stokes equations. In either case, to study the dynamics+/-= g(x) on R6 or to solve h(x)=0 in R6 would be a highly nontrivial taskmwere it notfor the symmetries in the Taylor problem. Both reduction procedures can be performedso as to respect these symmetries. Therefore, g and h will commute with an action of

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the symmetry group O(2)xS as we explain below. This places considerable restric-tions on the form that g and h may take. When, as here, we are studying only steadyand periodic states, it is sufficient to use the simpler Lyapunov-Schmidt reduction.This is our approach. For a complete study of the dynamics, the same restrictions’ onthe form of g will be true provided a smooth center manifold exists. (It is plausible thatthe symmetry might imply this, but we have not attempted to address this issue here.)

In this paper we give an explicit representation for all smooth mappings thatcommute with this action of 0(2)S1. We use the symmetries to show how to solve theequation h 0 (in the Lyapunov-Schmidt interpretation), and to determine (in mostinstances) the signs of the eigenvalues of the 6 6 Jacobian matrix dh[h__ 0. In particu-lar we compute these eigenvalues for the solutions corresponding to wavy vortices.

In this respect our results resemble those of a recent paper of Chossat and Iooss[1984]. However, instead of working on the six-dimensional kernel, Chossat and Ioosstrack the bifurcations step by step using the symmetry in the primary bifurcation toanalyze the possible types of symmetry-breaking at secondary bifurcations, in terms ofthe linearized eigenfunctions. The types of solution that they find can all be expressedas combinations of the six linearized eigenfunctions that make up the six-dimensionalkernel; but no reduction to R6 is used explicitly. Thus, although the various pieces ofthe bifurcation diagram are studied, their overall arrangement (and consistency) is not.

In our approach group theory is used to provide a coherent framework thatorganizes the analysis and in particular the computation of stabilities, leading to moredetailed results. In particular we confirm, in our setting, a conjecture made by Chossatand Iooss [1984] about tertiary bifurcation to modulated wavy vortices. We show that(with suitable parameter values) the branch of wavy vortices loses stability by a torusbifurcation. In experiments this transition is observed, the new state being calledmodulated wavy vortices. See Rand [1982], Gorman, Swinney, and Rand [1981], Shaw etal. [1982].

The analysis of the simplest (nondegenerate) O(2)Sl-symmetric bifurcationproblems on the six-dimensional kernel leads to a picture that includes branchescorresponding to a variety of the observed flows: Couette, vortices, wavy vortices,twisted vortices, spiral cells, modulated wavy vortices, wavy spirals and an unstableflow found numerically by DiPrima and Grannick [1971] which they call the "non-axisymmetric simple mode." By "correspond" we mean that the, solutions we find onthe six-dimensional kernel appear to have the same symmetries as the experimentallydetermined states. As we indicated in the introduction, it is further possible to chooseparameters in the model to mimic the observed transition sequences when the outercylinder is held fixed, and also in the counterrotating case.

DiPrima, Eagles, and Sijbrand [1984] are currently making numerical calculationsof certain of the Taylor coefficients of the vector field g obtained by a center manifoldreduction. These or similar numerical results should make it possible to determine towhat extent the six-dimensional model reflects the expected transitions in the Taylorproblem, at least in the counterrotating case.

2. Symmetries on the six-dimensional kernel. Symmetries are introduced in theTaylor problem in three distinct ways:

(1) by the experimental apparatus,(2) by the mathematical idealization,(3) by the mathematical analysis.

Since each of these ways introduces a circle group of symmetries the result may seemconfusing at first. However, these symmetries do affect the mathematically determined

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solutions and, moreover, seem to be present in the experimentally determined states.The symmetries arising through the apparatus would appear to be the most

natural. All formulations of the Taylor problem are invariant under rotation in theazimuthal plane, a plane perpendicular to the cylindrical axis. Rotation through O inthis plane moves one fluid state to another. We denote these symmetries by SO(2).

Next we discuss the symmetries introduced by the mathematical idealization. Inthe experiments, when vortex flow is observed these vortices tend to have squarecross-sections; that is, the height of each vortex is approximately equal to the distancebetween the cylinders. As a result, in an apparatus whose cylinder length is longcompared with the distance between the cylinders, many vortices form at the initialbifurcation. Moreover, the vortex flow appears to be invariant under translation alongthe cylindrical axis by two band-widths, at least away from the ends of the cylinder. (Inthe cross-sectional regions vortex flow alternates between clockwise and counterclock-wise.)

Thus in the mathematical idealization we assume that the cylinders have infinitelength and look only for periodic solutions of period equal to two band-widths. As aresult, the Navier-Stokes equations are invariant under both translations along thecylindrical axis and reflection of the cylinder through the azimuthal plane. Periodicityimplies that translation by two band-widths acts as the identity. Thus the effectiveaction of this group is by the (compact) group 0(2).

Finally, we consider a circle group of symmetries which is introduced into thisproblem by the technique we use to analyze the bifurcation structure. We use aLyapunov-Schmidt reduction to determine time-periodic solutions of the Navier-Stokesequations which lie near Couette flow and the parameter values yielding the six-dimen-sional kernel. The circle group S1, acting by change of phase on periodic functions,introduces symmetries into this problem. The addition of these S symmetries by theLyapunov-Schmidt procedure, to the symmetries mentioned above, is described inSattinger [1983] and Golubitsky and Stewart [1985].

We summarize our discussion here as follows. The full group of symmetries of theTaylor problem on the six-dimensional kernel is:

(2.1) 0(2) S0(2) S

where 0(2) acts by translation and flipping along the cylindrical axis, SO(2) acts byrotation of the azimuthal plane and S acts by change of phase of periodic solutions.For simplicity of notation we assume that the period of the cylindrical translations is2 rr and that the period of patterns around the cylinder (in the azimuthal plane) is also2rr. In particular, rotation of the cylinder by half a period is rr SO(2). Moreover, weassume that solutions are 2 r-periodic in time.

These assumptions do not affect the group-theoretic formulation of the problem,or its analysis; but they must be correctly interpreted in connection with the observedflows. Since the situation is potentially confusing, a few clarifying remarks may be inorder. There is no problem in arranging period 2 rr for translations: we merely scale thedistance along the axis. For periodic solutions in the azimuthal direction a little morecaution is required. For example, it is commonly observed in experiments that wavyvortex solutions may appear with wave numbers 3 or 4 (say); that is, with 3 or 4complete periods relative to a single turn of the cylinder. Provided only one such modeis present, we may scale the azimuthal angle to "factor out" this additional periodicity.The angle 2r then represents one period (2rr/3 or 2r/4 on the physical cylinder). Ingroup-theoretic terms, an action of SO(2) for which 0 SO(2) produces a rotation by

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kO, k an integer, can be viewed as the standard (k= 1) action of SO(2)/Zk and thisgroup may be identified with S0(2).

On the six-dimensional kernel, only one such periodic mode occurs, and thisprocedure may be followed. If two modes with different wave numbers occur, it wouldbe necessary to make S0(2) act by kO and 10 (where k, are the respective wavenum-bers) on the corresponding spaces of eigenfunctions and to carry out the analysis forthe appropriate action of 0(2) S0(2) Sx. See Chossat [1985].

3. Observed solutions and their symmetries. In the experiments a number of pre-chaotic states are observed. In this section we discuss the symmetries of each of thefollowing states: Couette flow, Taylor vortices, wavy vortices, spiral cells, and twistedvortices.

Both the Couette and vortex flows are time-independent. As noted above, vortexflow produces bands along which the flow is in the azimuthal plane. See Fig. 3.1(a).

When fl0__<0, vortex flow loses stability, and a time periodic state called wavyvortices appears. See Fig. 3.1(b). This periodic flow has the special form of rotatingwaves. More precisely, the solution u(t) is a rotating wave if u(t + 0) RoU(t) where R 0

denotes rotation by angle 0 in the azimuthal plane. We shall see in {}4 that all periodicsolutions obtained from the six-dimensional kernel must be rotating waves.

When f0 fl, wavy vortex solutions lose stability, and a new quasi-periodic solu-tion with two independent frequencies appears. This new state is called modulated wavyvortices. It is interesting to observe, at this point, how the modulated wavy vortexsolution might be detected by our proposed method using a Lyapunov-Schmidt reduc-tion. The idea is to compute the Floquet exponents along the wavy vortex branch ofsolutions and show that certain of these exponents cross the imaginary axis. Then applythe Sacker-Neimark torus bifurcation theorem to conclude the existence of quasiperi-odic solutions. We show in 7 that this scenario is possible. A similar remark holds foridentifying wavy spiral states when fl0 < 0.

We note that the actual transition to chaos cannot be explained by our analysis.Nevertheless, chaotic behavior may be present in our model and this point deservesfurther investigation.

We also note here that in the corotating and counterrotating Taylor problemssolutions with different planforms are observed. For example, in the corotating caseAndereck, Dickman and Swinney [1983] have observed twisted vortices. See Fig. 3.1 (c).In the strongly counterrotating case, Couette flow loses stability to a helicoidal patterncalled spiral cells, which are time-periodic rotating waves. See Fig. 3.1(d).

We can distinguish each of the states described above by their isotropy subgroups;that is, by the subgroup of (2.1) which leaves the given state invariant. In Table 3.1 welist the isotropy subgroups for each fluid state described above.

We now discuss the entries in Table 3.1. The steady-state solutions are invariantunder change of phase ($1); the periodic solutions are all rotating waves .and areinvariant under A since a change of phase may be compensated for by rotating thecylinder. Couette flow is invariant under all symmetries. Taylor vortices are invariantunder all rotations (SO(2)) and the flip along the cylindrical axis r. We denote byZ2(r) the two-element group generated by r.

Isotropy subgroups for the periodic solutions are obtained as follows. The helicalstate, spiral cells, is invariant under SO(2) since a translation along the cylinder axismay be compensated for by a rotation of the cylinder. Next observe that wavy vorticesare invariant under the group element obtained by composing the flip (r) with rotation

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(a) Taylor vortices (Ro= 1164, Ri= 1161). (b) Modulated wavy vortices Ro= 100, Ri= 350).In a still photograph wavy vortices and modulatedwavy vortices have a similar appearance.

(c) Twisted vortices R 721, Ri= 1,040). (d) Spiral cells R 295, R 237).

FIG. 3.1. Observed flows in the Taylor experiment. Reynolds numbers for the inner (Ri) and outer (Ro).ylinders. Photographs kindly supplied by Harry Swinney and Randy Tagg. Similar photographs will appear inAndereck, Liu and Swinney [1985].

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TABLE 3.1Isotropy subgroups of observedfluid states.

State Isotropy subgroup

Couette flowTaylor vorticesWavy vorticesTwisted vortices

Spiral cells

0(2)S0(2)SZ2 Ic X SO(2) X SZ_(x,r)AZ2(/)Aso(2)xzx

a {(O, -O)SO(2)XS0(2) is the flip z z along the cylindrical axisSO(2) is rotation of the azimuthal plane by one-half period

SO(2) (+, k) 0(2) SO(2)

of the cylinder by half a period r SO(2). Finally, twisted vortices are invariant underthe flip .

It is worth noting that the first three bifurcations in the standard Taylor problem(f0 0) break symmetry in a simple way. Couette flow to Taylor vortices breaks thetranslational symmetries; Taylor vortices to waw vortices breaks the rotational symme-tries (SO(2)); and wavy vortices to modulated wavy vortices breaks the rotating wavesymmetries (A).

4. Group theory and the six-dimensional kernel. In this section, we answer fourquestions:

(1) What is the exact form of the six-dimensional kernel?(2) What is the action of the symmetries of the Taylor problem on this kernel?(3) What is the form of the reduced mapping h, obtained by the Lyapunov-Schmidt

procedure?(4) What are the possible isotropy subgroups of points in the six-dimensional

kernel?We answer the first question by referring to DiPrima and Sijbrand [1982]. Let

rl Ri/Ro be the ratio of the radii of the inner and outer cylinders. We quote:

Thus, for example, for /=0.95 and f0/f -0.73976, Couette flow issimultaneously unstable to an axisymmetric disturbance with wave numbers(X,m)=(3.482,0) and a nonaxisymmetric disturbance with wavenumbers(X,m)=(3.482,1). We also note that...there are 6 critical modes with axial(Z) and azimuthal (19) dependence as follows:

(4.1) cos)kZ, sin X Z, e +- io cos()kZ), e ---iO sin()Z).

The action of the translations in 0(2) on the eigenfunctions in (4.1) is generated bytranslations of the axial (angle) Z and the flip (x) which acts by Z- -Z. Rotations inthe azimuthal plane act by translations in . (We have omitted the radial dependenceof the eigenfunctions here as the group 0(2)XS0(2) acts trivially in the radial direc-tion.) Observe that the resulting action of 0(2)xS0(2) on the six-dimensional spacegenerated by the eigenfunctions in (4.1) leads to the following equivalent action. Weidentify the six-dimensional kernel with

(4.2) V=a2 (R2(R) C)

and let elements of 0(2) act on R2 in the standard way and elements of SO(2) act by

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multiplication on C. That is

(0, )(v, w(R) z)= (Roy, ( Row)(R) (ei+z))where Ro is the usual rotation of R2 through the angle 0. Similarly, the flip x acts by(Kv, Kw (R) z), where K is the matrix (l). The action of the phase shifts S in V turnsout to be identical with the action of SO(2) on V. This fact can be verified by directcomputation. First observe that the R2 summand in Vis spanned by {cos(XZ), sin(XZ)},a steady-state kernel. As such, S acts trivially on R2. Next, observe that S commuteswith 0(2)SO(2) and hence the actions of S and 0(2)S0(2) on R2(R) C commute.Since the only matrices acting on R-(R)C commuting with 0(2)xS0(2) are scalarmultiples of matrices in SO(2) (see Golubitsky and Stewart [1985, Lemma 3.2]), itfollows that the elements of S act in a fashion identical to elements of SO(2). Withoutloss of generality, we may identify the actions of SO(2) and S1.

One consequence of this identification is that the subgroup A= {(q,-q)SO(2) x S } is in the isotropy subgroup of every element in V. Thus, we have proved"

LEMMA 4.1. Every periodic solution found in the six-dimensional kernel is a rotatingwave.

A second consequence of the identification of the actions of SO(2) and S is thesimple form that the reduction bifurcation equation h" V V, obtained via aLyapunov-Schmidt reduction, must take. (The function h depends on a number ofextra parameters, 20 for example. We suppress this dependence here.) Let the purelyimaginary eigenvalues of the linearized Navier-Stokes equations be + 0i. For simplic-ity, use a scaling argument to assume = 1. Then the idea behind theLyapunov-Schmidt reduction is to look for small amplitude periodic solutions ofperiod near 2 r. One does this by rescaling time in the original equation by a perturbedperiod parameter and looking for precisely 2r-periodic solutions to the scaledequations. What results, after appropriate applications of the implicit function theorem,is a reduction equation

where h: VR V is smooth and commutes with 0(2)S0(2)S. We claim that wemay assume that the dependence of h on is particularly simple. In fact,

(4.3) h(v,)=g(v)-(1

where J is the matrix form of the action by r/2 S on V.To verify this claim, suppose for the moment that there exists a smooth center

manifold. Let g(v) be the reduction vector field on that center manifold. It was provedin Golubitsky and Stewart [1985] that if the Lyapunov-Schmidt reduction is applied tog, introducing r, then the resulting function h has exactly the form (4.3). This fact relieson having the spatial symmetries SO(2) identified with the temporal symmetries Sa.

If we perform the Lyapunov-Schmidt reduction directly from the Navier-Stokesequations, then the reduced function has the same form as (4.3), at least to first order inr. In any case, the form (4.3) is used later only to solve certain equations forexplicitly. If higher order terms are present, then these equations may be solvedimplicitly, which is sufficient for our purposes. Therefore we lose nothing by workingwith h in the form (4.3). Moreover, we note that g commutes with the action of0(2)S0(2)S on V and may be identified with the mapping on V obtained by acenter manifold reduction, at least up to any finite order in its Taylor expansion.

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For the remainder of this section we describe precisely the form that mappings gwhich commute with 0(2)S0(2)S must have. Note that a third consequence ofidentifying the actions of SO(2) and S is that at this stage we may ignore one of them.Henceforth, we assume that

(4.4) r=o(2)xs’is our group of symmetries and turn attention to the action of F on V.

At this point we choose coordinates on V. First write V= V V2 where V1 R2 Cand V2= 112 (R) C---M(2, R), the space of 2. 2 matrices with real entries. Thus elementsof V have the form

(4.5) (z,A)where

z=x+iyC and A=( a b)M(2 R)c d

In these coordinates the group action of F O(2)S on (z,A) is defined as follows:

(4.6) (0, q)(z,A)= (eiz,RoAR)where

Ro=(CosO sin0)sin 0 cos 0

is the rotation matrix. See Golubitsky and Stewart [1985, 3] for more detail.We now answer the third question by describing in detail the invariant functions

and the equivariant mappings corresponding to this group action. (Recall that :V V is equivariant if d(v)=,ld(v) for all 3,F, v V.) Proofs are found in theAppendix.

PROPOSITION 4.2. Let q: VR be a smooth function defined in a neighborhood of theorigin which is invariant with respect to the action of F in (4.6). Then there exists a smoothfunction h: R R defined near 0 such that

where

(v)=h(fl,N,82,7,o)

fl= z= x2 +y 2,N=a2+b2+c2+d2,=detA,, Re(z2),o= ii Im( z2)

and

=a2 + b2-c2-d 2 + 2i(ac+ bd).

THEOREM 4.3. Let b: V- V be a smooth F-equivariant mapping defined near O.Then there exist F-invariant functions

p,q,r,s,p,p2, Q1, Q2, Q3, Q4,R1,R2,R3,R4,M3,M4

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such that

(4.8) ( 4

b(z,A) pz + qiz +r+si, E (SJK+ TL)j=l

where

and

KI=( ac(-cEl= a

d d c -c -d

b a a b L4__ ( -db

(4.10)

S1--p1,S2=p -,Tl=Rl+Ollm(z2),T=R28+OIm(z2),S 0 Re(-) +R38 Im() +M36 Im(2),$4= Q4Re(2) + R4t Im() -[- M4i Im(2 ),T Q3 Im(2) +R38 Re() +M31 Re(22),T4= Q4 Im(:z) +R4 Re() -1- M41 Re(2).

a),

We shall exploit the form of q, in (4.8) to solve explicitly the reduced bifurcationequation g 0. In order to understand what types of solutions one may find in g 0 weanswer the fourth question of this section. By determining, up to conjugacy, the set ofall isotropy subgroups of elements in V, we determine the symmetries that possiblesolutions to the Navier-Stokes equations found by reducing to V may have.

The lattice (of conjugacy classes) of isotropy subgroups for I’ acting on V is givenin Table 4.1. Containment of one conjugacy class in another is indicated by arrows. InTable 4.2 we list these isotropy subgroups along with the states in the Taylor problemwhich have those symmetries. We use the notation Z2 to indicate a two-element groupand Z2(a) to indicate the two-element group generated by F.

We emphasize that the containments in Table 4.1 are of conjugacy classes. Forexample, Z(xr,r) is not contained in Z2(/c)xS1. However, Z(xr,r) is conjugate toZ2 ( x, r ) which is contained in Z(x) x St.

TABLE 4.1Lattice of conjugacy classes of isotropy subgroups of F acting on V.

/, o(2)xs

z_ () s’ " so()

( z(,)z ) z(,) .,,

Note: Z is generated by , (,), (,).

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TABLE 4.2The symmetries associated with observedfluid states.

Isotropy subgroup Solution type

O(2)SZ2()xSso(2)

Z()

Couette flowTaylor vortices

spiral cellswavy vorticestwisted vortices

We derive the lattice pictured in Table 4.1 by first considering orbit representa-tives. Begin by considering the action of 0(2) S on R2 (R) C M(2, R). Let A be a 2 2matrix. As shown in Golubitsky and Stewart [1985, {}7], we can choose an element ofO(2)xS so that A is conjugated to the diagonal matrix () where a>__ d>__0. It is theneasy to show that there are four types of orbits as shown in Table 4.3.

TABLE 4.3

Orbit representatives of 0(2)S acting on M(2, R)

Orbit representative Isotropy subgroup

0

0,a>0

(a O)’a>O0a

0,a>d>0

o(2)xs

zi

SO(2)

Z2 (gr, qT"

Having put the matrices in M(2,R) into normal form, we now use the isotropysubgroups of these matrices to conjugate the elements z C-= R2. In this way we obtainrepresentatives for all the orbits of 0(2)XS acting on R2 (R2(R) C). These results aresummarized in Table 4.4.

TABLE 4.4

Orbit representatives of 0(2)S acting on R (9 (R (R) C).Orbit representative (z, A) Isotropy subgroup

0(2)SZ2(K)S

z

SO(2)

z2(,)

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TABLE 4.5

Fixed-point subspaces of 0(2)S acting on I/.

0(2)S

Isotropy subgroup Fixed-point subspace Dimension

Zz()xS

zso(2)

Z:(r,)

Z(,)

(x,0)

(0,(;

In the last table of this section we present the fixed-point subspaces of the variousisotropy subgroups. More precisely, let c F be a subgroup. Define

(4.11) Vy= (ve Vlov=v for allo2:}.

Observe that if " V V commutes with F, then maps V to itself (see Golubitskyand Stewart [1985, (1.6)]).

5. Branching and stability. Let h (z,A, X, ) be the mapping on the six-dimensionalkernel obtained via the Lyapunov-Schmidt reduction. Note that h depends explicitlyon the bifurcation parameter and the perturbed period z. In addition, we know that hcommutes with the symmetries in the Taylor problem. We shall use the consequences ofthis fact to explain how to compute both the solutions to h- 0 and the eigenvalues ofthe 6 6 Jacobian matrix dh along branches of solutions to h- 0. One consequence ofthe 0(2)S0(2)S symmetries is that the eigenvalues of dh determine the orbitalasymptotic stability of solutions.

Let us be more precise. Recall the form of h in (4.3) with its simple --dependence,namely

h(z,A ,z)=g(z ,X)-(l+’)(O,(-bd a))cwhere A =(,a.). In deriving this form we note that if the Navier-Stokes equationsadmit a smooth center manifold then h will have exactly the form (5.1), where g is thereduced vector field on that center manifold. We proved in Golubitsky and Stewart[1985, Thm. 8.2] that the eigenvalues of dh determine the orbital asymptotic stability ofsolutions to the vector field g on the center manifold. Moreover the center manifoldreduction implies that the stabilities of solutions to the vector field g are the same as thestabilities of the corresponding solutions to the Navier-Stokes equations.

If there should not exist a smooth center manifold (which we doubt) then we arecomputing the correct stabilities for g, accurate to any finite order, by using theeigenvalues of (dh) h--0-

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SYMMETRY AND .STABILITY IN TAYLOR-COUETTE FLOW 263

We now describe how to compute the eigenvalues of dh. Recall that (4.8) restrictsthe form of g in (5.1) to:

( 4

(5.2) g(z,A,,)= pz + qiSz +r+ sii2, E SJKj+ TJLjj=l

where p,q,r,s and the coefficients p1,...,M4 appearing in (4.11) are invariant func-tions, hence functions of the five variables fl, N,8 2, ,, o defined in (4.7) and . More-over, since h is obtained via the Lyapunov-Schmidt reduction, the linear terms mustvanish. Hence

(5.3) p(0)=0, pI(0)=0, e2(0)=l.

Equivariance shows that to solve the equations h 0 we need only evaluate h ontypical orbit representatives. The resulting equations are listed in Table 5.1. See Table4.4 for the list of orbit representatives and their isotropy subgroups.

Remarks. (i) The equations involving r serve only to determine the perturbedperiod of the associated periodic solution. Note that (5.1) allows us to eliminate r bysolving these equations explicitly (or implicitly if there does not exist a smooth centermanifold, see 4 above).

(ii) Observe that r is indeterminate on the Z2(x)S branch, which is to beexpected since the bifurcation is to a "steady state."

(iii) Observe that the theoretical basis of our explicit calculations is given by (4.10):fixed-point subspaces V are mapped to themselves by equivariant mappings. There-fore we may restrict h to V and seek solutions to h V= O, considering each isotropysubgroup Y in turn.

Table 5.1 also lists the coefficients that determine the signs of the (real parts ofthe) eigenvalues of dh. We consider the branching equations briefly first, and thendescribe in more detail the eigenvalue calculations.

By writing (4.9) in coordinates we obtain

a b))=(X,y,(5.4) h(x’Y’(c d

where

(a) X=px-qSy+ rxRe(f)+rylm(f)+sSyRe(f)-sSxlm(f ),(b) Y=py + qSx ry Re(’) rx Im(’) +sSx Re(’) +sSy Im(’),(C) A--(sX+S3)a+(-S2-S4)bnt-(-Tl+ T3)c+(T2-T4)d,(d) B--(S2+S4)aq-(sX-t-S3)b+(-T2-t-T4)c+(-TI+ T3)d,(e) C--(TI-t-T3)a+(-T2-T4)b+(S1-S3)c-t-(-S2nt-S4)d,(f) D=(T2+T4)a+(TX+T3)b-t-(S2-S4)c+(S1-S3)d.

The branching equations always take the form X= Y A B C D 0, evaluatedon the appropriate orbit representative. The entries in the table follow readily. How-ever, a few comments should be made regarding the last four entries of "unknown"type.

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TABLE 5.1

Branching equations and eigenvalues for solutions with given symmetry.

Solution type"Isotropy;orbit

Couette flow0(2)S(0,0)

Taylor vorticesZ2(1) XS(x,0)

unknown

spiral cells

S0(2)

Branching equations(to be evaluated atorbit representativeshown)

None

p=0

at’ (x-, 0, 0, 0, 0, X) Ill

p1 0p2=l+z

at: (0, a2, 0, 0, 0, k

p1 + a2R =0

p2 a2R 1 + r

at: (0, 2a2, a4,0,O,k)

Signs ofeigenvalues

Pa _+ i(p2 1 -r)

0

(p1 + x2Q3)+i(Pp#2_ l_r + x2Q4)(p2_ x2Q3)+_i(p2_ l-r- x2Q4)

0p+ ra

p-- ra

PIN + a2pR + a2R4

oP _+_ iqa

2P + R + a2(p2 + 2R2N)+a4R22(R2+2a2Ra)+i(R +2a2R4)

Multiplicity

2,2

11

1,11,1

11

1

1,1

1

1,1

wavy vorticesZ2(r,r)

twisted vorticesZ2()

((ax 0

p-a2r=Opa _y2Q3 =0p2 _y2Q4._ 1 + "

at: (y2,a2,0,- a2y2,0,k)

R2a2 + 2Q3y2 + ay2Q+a4R4-a2y2M4

Aydet =p/ pX (PN r)(PA Q3)+

trace=py +pa +

unknownZ2(r,r)

o))p+a2r=Op1 + x2Q3__Op2 + x2Q4 1 + r

0

R2a2-2Q3x

ax2Qt + a4R4+ a2x2M4

at: x a 2, O, a2x 2, O, ,

R2+(a2+d2)R4=Oplus others

at: (0, a + d a2d O, O, ))

X Xa ][2]A A,det =pt P-(pN_+ r)(P Q3)+

trace=p/x +P}a +not computed

1,1

1,1

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TABLE 5.1 (continued)

unknown1

Oa))unknown1

(x+iy,( a0

unknown1

q=0plus others

at: (x2,2a2,a4,0,O,h)

r=0plus others

at: (x +y2,a2,0,(x2-y2)a2,0,)X=Y=A=B=C=D=O

at: x +y2, a + d, a2dx -yZ)(a2 d),2xyad(a d2), k)

not computed

not computed

not computed

Notes: [1] The equations must be evaluated at (fl, N,82,’,o,)) where these in turn are evaluated on anorbit representative to yield the form stated.

[2] The remaining eigenvalues are those of the specified 2 2 matrix. Its determinant and trace areshown to lowest order (omitting positive factors) to determine their signs.

When the isotropy group is Z2(r, r) we take two of the equations, namely A --0D, which reduce to:

(aid S3)a+(T2- T4)d= 0,

(T2+ T4)a+(S1-S3)d=O.Now observe that $1+ S and T2+ T4 have a factor d. Divide this out and subtract.The result has a factor (a-- d2), and the entry in the table follows. We do not requirethe remaining equations, because an appeal to nondegeneracy (6) now rules out thiscase.

For the next two cases, the equations X= Y= 0 lead, among other things, to thelisted equation, which is also ruled out. by nondegeneracy. The final case is extremelycomplicated and it remains possible that such a branch might occur: see 7 for furtherdiscussion.

The computation of the eigenvalues, particularly those along the Z2(x) andZ2(xr,) branches, is the most difficult part of this section. This computation isfacilitated by the use of several results in Golubitsky and Stewart [1985, 8b]. The firstis that along a solution branch (v0, 0, %) with isotropy subgroup Z, the derivative

(dh)v0,x0, commutes with Z. This implies (Lemma 8.4 of that paper) that (dh)vo,X0,leaves invariant the subspaces W of V formed by adding together all subspaces of V onwhich Z acts by a fixed irreducible representation. We use the Wj. to put dh into blockdiagonal form.

The second fact is that (dh)o,x, vanishes on all vectors tangent to the orbit of vunder the action of 0(2) S1. These null-vectors are given by:

(5.6)(a)

(b)

-1

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We now outline the explicit computation of the eigenvalues listed in Table 5.1.(a) Z2 S (Taylor vortices). Decompose V= R2 (9 (R- (R) C) into irreducibles for

Z S1. We get V= Wo (9 W (9 W (9 W where

a b((0, (0 0))),WI=((iyO)} W3=((0,(0 0)))c d

The actions are given by

-1 11 R

which are distinct irreducible actions. Therefore, dh leaves each invariant. Let(j= dhlWj., so that dh has the block form:

y ab cd

0 0 0(I)

0 (I)2 0

(I)

We evaluate the (I)j at the orbit representative (x, 0), with the following results.(I)o= Xx=p +PxX=pxx since p=0 on this branch by Table 5.1. Now x > 0 so we

can divide it out.

(l Yy=p +pyy=O,

[aa hb]=[ sl+s3(= Ba Bb S- + S4 -S2-$4] [ pI+x2QSI + S p- l -,r + x2Q4

-(P-l-’+x2Q4)

Since a matrix ( -) has eigenvalues a + ifl, the entry in the table follows. Similarly

De Da $2-S4 S1- S =[ pI-x2Q3p2 -1- ,r x2Q4

( p2-1- "r- x2Q4 )x2Q

The entries for vortices in Table 5.1 follow. Note that (I)2 and (1) have to be scalarmultiples ( -,) of rotations since dh commutes with $1; direct calculation confirmsthis.

(b) Z (unknown). We use the same decomposition V= Wo (9 W1 (9 W2 (9 W3 as in(a). The actions are:

Wo Wa W2 W

x 1 -1 1 -1(r, rr) -1 -1 1 1

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So the irreducible components on these spaces are distinct. Therefore dh leaves eachinvariant: set j= dhlW. We have

d#o =p + ra 2, b =p ra 2.

Now dh has one eigenvalue 0 on each of W2 and W3, so the remaining eigenvalues areTr 2, Tr3. Use (5.4) here. These are computed as follows.

Tr*2=Aa+Ba=(SI + S)+(Sa + S3a )a+(Sl + $3)+(Sa2+ S)asince b c d 0 on the orbit. The branching equations show that S + S 0, so thesign is given by S + S +S+ S. Now on the orbit we have z 0, 8 0, " a 2, 8 0,8b=0. So by (4.11) we compute this as pl+p. But Nb=O on the orbit, so thisbecomes P. 2a. Dividing by 2a > 0 gives the table entry. (From now on, we omit suchdetails from the calculation.)

Similarly,

Trd3=Cc+Dd=Sl-S2+(Tcl+ Zc3)a+(Sx-S3)W(Zf + Tff )a.But S -[- S --0 on this branch, so this is

4SI + a( TI. + Tc3 + T+ T) a:ZRg- + a4R4.

(c) SO(2) (spiral cells). Let

a b }}These are irreducible under S6(2); and (0,-0) S6(2) acts on W0 as e i, on W asthe identity, and on W2 as e2 (see Golubitsky and Stewart [1985, (10.5)]). Hence theWj. are invariant under dh. Let j= dhlWJ. as usual. We compute

which has the form (-) required to commute with SO(2). Now (I) has one zeroeigenvalue on W, so the other is

TrdPl =Aa +Ad+ Bb- B

by Golubitsky and Stewart [1985, (10.11)]. We compute this on the orbit (0, (g 0a)). Theresult is

TRY1= 2a(Sa + Sa + Ta- Ta4+ S + Sd4+ Td2- Tff).

In deriving this, note that S + S + T3_ T4__ 0 by the branching equations. Also,the b- and c-derivatives of the invariants are equal, so the b- and c-derivative termscancel. The a- and d-derivatives of/3, N, , ,, 0 are equal, whereas a 2a -’d. Hencethe only terms that remain, on dividing out positive factors, are

2pIN+ R2+ a(P2 + 2RN)+ a4R22.

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The matrix of (I)2 can be computed as

Aa-Ad Ab+Ac]Ba-Ba Bb+B

and must be of the form (-) by Golubitsky and Stewart [1985, 10]. Much as above,we find that

Aa-Aa= 2a2(R2 + 2a-R4),Ab+Ac= -2a2(Rl+2a2R4),

as required for the entry in the table.(d) Zz(xr, r) (waoy oortices). We decompose V= Wo W1 where

W0 {y, a, b) + 1 eigenspace,

W (x, c, d) 1 eigenspace,

and take a basis in the order

y,a,b;x,c,d.

Let 9 dh[W. The null-vectors for the two zero eigenvalues of dh may be found from(5.4) and are

0 0

0 and _0y0 a0 0

with respect to this basis, when evaluated on the orbit. So column b of dh is zero andcolumns x, e are linearly dependent. Direct calculation yields Cx Dx=0, whence bylinear dependence C,. Dc= 0. So dh is of the form

y a b x c dy 0 0 0 0a 0 0 0 0b 0 0 0 0

x 0 0 0c 0 0 0d 0 0 0

0 00 0

The eigenvalues of 2 are therefore

Xx=2ra 2,

Dd= aZR2 + 2yZQ + ayZQ+ a4R4 aZyZM4,

and those of I) are given by

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Now

Yy=p +pyy ra 2 ryya - =pyy rya2yYa=pay-(aZra+2ar)y,Ay=(Sy+S3y)a,Aa=(SI+s3)o since $1 + $3 =0.

To evaluate the y- and a-derivatives, note that on the orbit,

Ny=O, fly=2y, (82)y=0, yy=-2ya2, oy=O,Na=2a, fl=0, (2)a--0 "ya’---2ay 2, Oa=O.

The y-derivatives introduce a factor y, the a-derivatives a factor a. So the matrix isof the form

[ 1e11Y2

el2aY

e21ay e22a2

for certain functions eij. We therefore evaluate the eij to lowest order. The result is

ellY

e21aY e2.a - (Sly+S3y)a (Sla+S3a)a

-[ 2pCy2 2(PN--r)ay

2(P-Q’)ay 2Pa2

The determinant and trace of therefore have the signs indicated in the table.(e) Z2(x) (twisted vortices). This is similar to (d). The decomposition into invariant

subspaces is now V= W0 Wx where

W0 (x, a, b) + 1 eigenspace for x,

W (y, c, d ) 1 eigenspace for x.

We take a basis for V in the order

x,a,b;y,c,d.

Then dh has block form, and we let dhlW .There are two zero eigenvalues of dh given by (5.4). In the basis above the

associated eigenvectors are

x 0 -y

ab ba c

--* 0and x

d

i dc ab

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Evaluation on the orbit representative (x, (g )) yields the eigenvectors

0 0

0 and 0x

0 a0 0

Therefore column b of dh is zero and columns y and c are linearly dependent. Puttingthe zeros in column b, we get

x

a

dq= b

Yc

d

x a b y c d

* * 0 0 0 0* * 0 0 0 0* * 0 0 0 0

0 0 00 0 00 0 0

Direct calculation, evaluating at y b c d 0, yields Cy 0, Dy 0, whence by lineardependence of columns y and c we also have Cc= Dc= 0. So

(I)1"-- 0 0 Cd0 0 Dd

This is triangular, so its eigenvalues are

O,Yy=p- a2r 2ra 2,

Dd= a2R2- 2x2Q ax2Q+ a4R4 4- a2x2M4,

using the branching equations.Since column b of 0 is zero, the eigenvalues of 0 are 0, together with those of

Now

Again the matrix is of the form

Xx =pxx + rxXa 2,

X=px+(a2G+2ar)x,Ax=(S+S)a,ao=(S +SJ)a.

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and we may retain only the lowest order terms in the fij. The result is

[2px22(PN+r)ax ]2(P-Q3)ax 2Pva2so the determinant and trace have the indicated signs.

Notice the "duality" between wavy and twisted vortices. This completes the verifi-cation of Table 5.1.

6. Nondegeneracy conditions. We now proceed to a detailed analysis of the solu-tions to the branching equations, and the signs of the real parts of the eigenvalues alongbranches, which determine (orbital asymptotic) stability. The main qualitative featuresof the bifurcation diagrams, and their associated stabilities, depend upon the signs of anumber of coefficients. We therefore impose appropriate nondegeneracy conditions:these coefficients should be nonzero.

Recall from 1 that in order to obtain the six-dimensional kernel we had to fix thespeed of counterrotation of the outer cylinder at some critical value f. At this speedwe found a two-dimensional eigenspace associated with zero eigenvalues, 112, coalescingwith a four-dimensional space associated with a pair of complex conjugate purelyimaginary eigenvalues, R2 (R) C. Thus, in order to model the effects of counterrotation inthe Taylor experiment, we must introduce a perturbation parameter a which will splitapart the bifurcations corresponding to R2 (vortices) and R2 (R) C (spiral cells and Z22).

We choose to do this by replacing p1 in (5.2) by a+P. If a <0, then thebifurcation to vortices occurs second (in the bifurcation parameter k= f;); if a > 0, itoccurs first. Thus, we may think of a as f]0-f]. See Fig. 6.1.

vortices

Couette

vortices spiral cells

g h

a<O a>O

FIG. 6.1. Schematic rendition of the effect of the perturbation a. The directions of the branches are chosen

arbitrarily and secondary branches haoe been suppressed.

The nondegeneracy conditions we impose on h are stated when a 0 and whenz=O,A =0.

In Table 6.1 we list sixteen nondegeneracy conditions; a F-equivariant bifurcation

TABLE 6.1Nondegeneracy conditions for h.

(a) Pt (h) R(b) Px (i) 2P + R(Cl) (P + Q3)Px PPB (j) 2(pNP-_Pvpx)-pxR(ca) (p_Q3)px_pp (k) pp-(p +Q3)(Pu+r

p1() P O) p-(P-Q )(e-r)(e) P (m) r

( P(p+r)-exP (n) Q3() P(p-r)-exP (o) q

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problem h is called nondegenerate if these sixteen expressions are nonzero for h. InTable 6.2 we list the lower order terms for each solution branch of h=0 and eacheigenvalue of dh along these branches. For nondegenerate h these lower order termsdetermine the direction of branching (super or subcritical) and the (orbital) asymptoticstability of each solution. In Table 6.1 we use the convention that all quantities are tobe evaluated at the origin. So, for example, PN means P(0,0,0, 0, 0, 0). The terms(a)-(n) in Table 6.2 refer to the corresponding expressions in Table 6.1.

TABLE 6.2Branching equations and eigenvalues to lowest order for nondegenerateproblems.

Couette flow(O(2)S1)(0, 0)

Taylor vortices

Z2 (/) XS

(x,0)

unknown

spiral cells

so(2)a 0))

wavy vortices

Z2 (to q’/’,

Branching equations

None

a lx2

-1h=’-[a + (d)a

x= _1___[(e)

a+(i)a

(c2) y2+ (g)a

(b)

(all !d a

Sign of real partof eigenvalues

Multiplicity

(b)Xa+(e)X

(e)(b)(e)

0(f)

a

(g)a+-a(d)(h)

0

(b) j) a2(e--- a+ -(i)-(h)

(m)

2(n) y2 4- (h)a

detY"

=(l)Aa

=(a)y2+(d)a

(b)

0

(a)

Oq-(Cl) X2

Oq-(C2) X2

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TABLE 6.2 (continued)

twisted vortices

o))

C1X

(f)a

b +-(- =a

a (ka2,_(a) +C1 1

tr[ x

-(m)

2(n)x +(h)a

det[ XxAx

Z r, r) No solutions by (h)

02Aa =(a)x2+(d)

No solutions by (m), (o)except perhaps on the orbit

*The terms (a)-(o) are defined in Table 6.1 and required, by assumption of nondegeneracy, to be nonzero.

In our analysis of the bifurcation diagrams and the asymptotic stability of theassociated solutions, we use only the equivariant form of the bifurcation equations andthe implicit function theorem. Moreover, in each appeal to the implicit function theo-rem we find a neighborhood of the origin in (z,A,?, a)-space on which its consequencesare valid. Since we use the implicit function theorem only finitely many times, all of ourconclusions hold simultaneously in some fixed neighborhood of (0,0,0,0) in(z,A,?,a)-space. This neighborhood does depend, however, on the particular valuesthat enter into the nondegeneracy conditions.

The computation of the entries in Table 6.2 may be completed in a routine fashionusing the entries in Table 5.1. We give the flavor of these computations by presentingthe results for wavy vortices.

The branching equations for Z2(Kq/" q?) are

p ( y2,a, 0 2, a2y 2, O,2 ) -a2r( y2,a2, O, a2y 2, O,X) =0,a + p(y2, a2, O,a2y 2, O, X)+y2Q3 (yZ,aZ, O, a2y, O, X) O.

See Table 5.1. (The third branching equation in that table is used only to eliminate -.)Expanding to lowest order, we have

0 =pt(0)y2+(Pu(O) --r (0))a 2 +px(0)2 +...,

O=a +P(0) y 2 + PN(O)a 2 + Px(O) 2- Q3(O) y2 + ....Using the implicit function theorem, we can solve for ) and y2 as a function of a 2 if

(o)( (o)), o.

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This is condition (c_) of Table 6.1. We then obtain

P(O)p#(O)- (PJ(O)- Q3(O))( pN(O)- r(O))a2+ ...,

px(O) Px(O)p(O)-P,(O)( plN (0) r (0)) a2+ ....

Using the entries in Table 6.1 along with some rearrangement of terms, we obtain theentry in Table 6.2.

7. Comparison with experiment. In this section we discuss how the above modelbifurcation problem(s) on the six-dimensional kernel compare with experimental ob-servations in the two main settings.

(1) Experiments by Andereck, Liu, and Swinney [1984] in the counterrotating case,including parameter values near a point at which the six-dimensional kernel ap-pears to occur.

(2) The standard "main sequence" of bifurcations in the case where the outer cylinderis held fixed:

Couette flow Taylor vortices wavy vortices modulated wavy vortices

We will show below that it is possible to make choices for the signs of thecoefficients that appear in Table 6.1 as nondegeneracy conditions, so that the resultingbifurcation sequences are in qualitative agreement with the experimentally observedbifurcation sequences. In the counterrotating case we have direct (numerical) evidencefor the existence of the six-dimensional kernel through the work of DiPrima andGrannick [1971]; no evidence for this six-dimensional kernel currently exists when theouter cylinder is held fixed. We hasten to add, however, that the existence of thesix-dimensional kernel is, because of symmetry, only a codimension one phenomenon;it should occur frequently in various forms of Taylor-Couette flow. We also note that,unfortunately, there are many different choices for the signs of the nondegeneracyconditions in Table 6.1 (over 10,000), so many different bifurcation sequences are

possible besides the ones we consider here. However, the possibilities are not totallyarbitrary, as we see below.

7.1. The counterrotating case. In a private communication, D. Andereck gave usthe (qualitative) form of the experimental results for counterrotating Taylor-Couetteflow, which have since appeared in Andereck, Liu, and Swinney [1984]. We presentthese results in Fig. 7.1. There are three features that deserve mention here.

(i) There is a critical speed of counterrotation f at which the primary bifurcationfrom Couette flow changes from Taylor vortices to spiral cells. This corresponds to thecritical speed of counterrotation found numerically by DiPrima and Grannick [1971].However, they presumably performed the calculations for values of the dimensions ofthe apparatus which differ from those used by Andereck, Liu and Swinney [1984].

(ii) In the weakly counterrotating case f0<f--this corresponds to the case

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(a > O) where our perturbation parameter a is positive--the observed bifurcation se-quence is"

Couette vortices wavy vortices

See Fig. 7.2.

20

weaklycounterrotating

( > o) ao > a

( < o) 0 < "stronglycounterrotating

Taylor

modulated

\ wav_ywaves

Couette flow -,vy spirals interpe.netrating

FIG. 7.1. Qualitative version of the experimental results of Andereck, Liu, and Swinney [1984] showingobserved transitions between stable states in the counterrotating Taylor-Couette system.

Couette

rtices

FIG. 7.2. Schematic bifurcation diagram when a > 0 corresponding to the observations of Andereck, Liu,and Swinney [1984]. (Secondary branches may or may not join other branches, depending on the values of the

coefficients.)

If the speed of the inner cylinder is increased further, then the wavy vortices losestability to another state which does not appear to correspond to any state in ourmodel.

(iii) In the strongly counterrotating case where f0 is slightly greater than f--thiscorresponds to our a < 0--the observed transition sequence is:

Couette spiral cells wavy spiral cells

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See Fig. 7.3.

Couette

rals "at*’spiral,,

g ,,cel.s

FIG. 7.3. Schematic bifurcation diagram when a < 0 corresponding to the observations of Andereck [1984].

In order to reproduce qualitatively the experimental results, we demand the following:

(a) Couette flow is stable for 2 << 0.(b) Vortices bifurcate supercritically and stably when a > 0.(c) There is a secondary bifurcation from vortices to wavy vortices when

a>0.(7.1) (d) Wavy vortices are supercritical and stable at the initial bifurcation from

vortices.(e) Spiral cells bifurcate supercritically and stably when a < 0.(f) Spiral cells lose stability to a Hopf bifurcation.

We claim that it is possible to choose signs for the nondegeneracy conditions inTable 6.1 so that each of the conditions in (7.1) is satisfied. Moreover, we claim thatwhen these nondegeneracy conditions are satisfied, it follows that:

(7.2) Any bifurcation from vortices to Taylor vortices occursafter the bifurcation from vortices to wavy vortices.

The assumptions in (7.1) correspond, in order, to the following nondegeneracyconditions (cf. Table 6.1):

(a) (e) <0, (b)<0.(b) (a)>0.(c) > 0.

(7.3)(d) (/)>0; (m)>0, (n)>0.(e) (i) >0, (h) <0.(f) (j)>0.

It is a simple matter to check that the conditions (7.3) are precisely the conditionsneeded to satisfy (7.1). The only point which requires comment is asymptotic stabilityof the wavy vortex branch. Observe that at the bifurcation from vortices to wa_vyvortices, a 0 and y =# 0. It follows from Table 6.2 that the wavy vortices are stable if

(m)>0, (n)>0, (/)>0, and(a)>0.However, (!) > 0 when the branch of wavy vortices is supercritical, and (a) > 0 hasalready been assumed in (7.3b). Observe that the branch of wavy vortices can lose

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stability if either

(7.4) (d)<0 or (h)<0.

If (d)< 0 then this branch will lose stability to a torus bifurcation. However,

(7.5) (d) ((i)- (h))/2,and the assumption that spiral cells are stable implies that (i)> 0, (h)< 0 (cf. (7.3e)) sothis possibility cannot occur. Nevertheless, (h)< 0 is satisfied, and wavy vortices maylose stability by a single real eigenvalue passing through zero. This observation verifies(7.2a). It is possible that a new solution branch with isotropy subgroup 1 will appear atthis bifurcation, but we have neither confirmed nor eliminated this possibility. SeeTable 5.1.

Finally, we verify (7.2b). The wavy vortex branch begins at )tw(a)/(c2) while abranch of twisted vortices would begin at )t=(a)/(c2). We compute sgn()/-w).Now

(7.6) sgn(X,-Xw)=sgn (cl) (c2)since (a) > 0 by (7.3b). However,

(c2)- (cx) 2Q3px 2(n )(b) > 0

using (7.3a, d). Hence (7.6) implies that t>w as claimed in (7.2b). Observe that it ispossible, under different circumstances, for twisted vortices to bifurcate supercriticallyand stably from vortices. Such a transition has been observed in the corotating case.See Andereck, Dickman and Swinney [1983].

7.2. The main sequence. Here we verify that the main sequence of bifurcationscan also occur in the six-dimensional model. This sequence of bifurcations is observedin experiments when the outer cylinder is held fixed. For this sequence to hold, we needCouette flow to lose stability first to vortices. This happens in our model when a > 0,and we concentrate on this case.

To obtain the main sequence, we need (7.1a, b, c, d) to hold. Of course, this ispossible precisely when the nondegeneracy conditions (7.3a, b, c, d) hold. If we wish toshow in this model that, in addition, the wavy vortex solutions lose stability to a torusbifurcation, then two complex conjugate eigenvalues must cross the imaginary axisalong the branch of wavy vortices. This can happen only if (d)< 0. Note that if (d)< 0then (7.5) implies that (7.3e) is not valid, and that spiral cells cannot be asymptoticallystable.

As we saw above, wavy vortices can lose stability by a real eigenvalue crossingthrough 0. However, this eventuality cannot occur if (h)> 0, which is possible, since wehave assumed nothing about (i). Thus, assuming

(d) <0, (h)>0leads to the main sequence. (See Fig. 7.4.) Other choices for the main sequence arepossible.

We conclude that both the main sequence and certain regimes in the experimentsof Andereck, Liu, and Swinney [1984] appear to be qualitatively consistent with oursix-dimensional model, for suitable values of the coefficients. (Note that aside from thestates discussed above, no other stable states occur except perhaps with isotropy group

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1, as mentioned.) Since the coefficients can in principle be computed byLyapunov-Schmidt reduction from the Navier-Stokes equations, further numericalwork should be able to provide a more stringent test. It would also be of interest todetermine, in terms of the physical parameters in the problem, the location of thecodimension one set of values at which the six-dimensional kernel occurs.

vortices

Couette 1

Vortices

FIG. 7.4. Schematic bifurcation diagram corresponding to (one occurrence of) the main sequence.

Appendix. Equivariant mappings on the six-dimensional kernel. Let V= R2 (R2 (R)

C) be the six-dimensional kernel described in [}4. Recall that I" 0(2) S acts on V by

( O, )( v, w(R) z)= (Roy, (Row) (R) (eiZ)).For computational purposes we choose coordinates by identifying the first R2 with C,and R (R) C with 2 x 2 matrices as described in 4, so that an element of V is written(z, a) where

zC, A=( a b)c d’a,b,c,dR.

Recall that a (smooth) function : R is incariant under F if

,()=,(), r, v,and a (smooth) mapping : V Vis equivariant if it commutes with F, that is

The aim of this appendix is to describe completely these invariant functions andequivariant mappings, as promised in 4 above. The main result, wch will yieldTheorem 4.3 when appropriate terms are collected together, is:

PROPOSITION A.1. (a) Every invariant function on V is of the

#r a smooth #nction h: R R, where

=Z=x2+y 2,N=a2+b2+c2+d 2,

(A.2) 82=(ad-bc)2,

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and

(A.4)

(A.3) ’= (a : + b:- c:z- d 2) + 2i(ac + bd ).(b) Every equivariant mapping, V V is of the form

d (z,A) ( pz + qiSz +r+siS,

P Re(H) +P2 Re(H2)+ Q11m( z 2g)lm(Hx) +eIm( zg)Im(H)+O Re(e n )+e4Re(e .4)+ RX8 Im(H)+R28 Im(H)+R38 Im(gH3) + R46 Im(gH4)

where

and

c+ ia d+ ib Ha= -c+ ia -d+ ib

d-ib c+ia H4= -c+ia

p,q,r,s,p1,p, 01, Q2, Q3, Q4,R1,R2,R3,R4,M3,M4are invariant functions.

In more abstract language, Proposition A.1 says that the ring of invariant func-tions is generated by fl, N,82,7,o; and that the module of equivariant mappings isgenerated over the invariants by the twelve mappings in (A.4), whose coefficients arep,q,r,s,P, .,M4. By standard results of Schwarz [1975] and Po6naru [1976] we mayassume q, and are polynomials when proving Proposition A.1.

The computation comes in two stages. First we compute the (polynomial) S-invariants and -equivariants; then we use this information and the O(2)-action toobtain the O(2)S-invariants and -equivariants. Since S acts trivially on z R, weneed consider only the action on A R2 (R) C. We take complex coordinates

z =a+ ib, z=c+ id.

Then we may identify R2 (R) C with C C, where S acts diagonally:

O(Zl,Z2)=(eizl,eiz2).LEMMA A.2. The real sl-invariants on C$C are generated by zl51, z252, Rez152,

Imz52. The Sl-equivariants are generated over the invariants by (z,O), (0,Zl), (z2,0),(0,z_), (ix, 0), (0, i), (i2,0), (0, i2).

Proof. These results (which generalize easily to S acting on C") are no doubt well-known, but for completeness we sketch a proof. The idea is first to find the complexinvariants and equivariants and then to read off the real ones.

Consider a C-valued polynomial functionDow

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Since eig=e-i, we can use sl-invariance to exclude all terms other than those forwhich

a-B+-=0.So p is a polynomial in__z121, z222, 1z2, and z122. If p is to be real in a,b,c,d, then wehave p =, so Aav=A/v. This leads to the real invariant generators stated.

For the equivariants, we consider a pair of functions p,p_ of the above form.Equivariance excludes all terms other than those for which

a-fl+7-3=l.This yields equivariant generators which are complex scalar multiples of (z1, 0), (z2, 0),(0,Zl), (0,z2). Taking real and imaginary parts, we obtain the stated real equivariantgenerators.

In () coordinates, we have the invariant generators

z,x=a2+b,._C2 2

(A.) zz +d

Re(z) ac + bd,

Im(z) be ad=

and the equivariant generators (a) ___>

(A.6) (a b)E2=(0 0)E3=(0 0)E4.__(C 0d)El= 0 0 a b c d 0

-b a 0) ET=( 0 0) E8=(-dEs=(0 0) E6=( 0b a cd

Note that there is a relation

(a 9- + b2)( c2 + d 2 ) (ac + bd )2 + (ad- bc)2.We are now ready for the:

Proof of Proposition A.1 (a). The calculations are easier if we use complex notation.Let

’= (a 2 + b2- C2- d 2) + 2i(ac + bd).Then every real-valued function of the four invariant generators can be written in termsof N=a2+bZ+c2+d 2, , , and 8. Note that N and 2 are O(2)-invariant. SeeGolubitsky and Stewart [1985, 9].

Since S acts trivially on z R2, we can write the sl-invariants on R2 (R2 C) inthe form

+(z,5,N,,,6).Under the O(2)-action these transform as follows:

z 8 N

x z2io -8 N

0 eiz e-i e2i e- N

(The expressions " and " are introduced because of this pleasant transformation behav-ior.)

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Since z, ’, N, and 2 are O(2)-invariant, we can write the general O(2)S1-

invariant in the form

(A.7)

where a, b,-.. ,h C[z,N, 8 2]. (Note" ’g= N-42 SO no ’g terms are required.) Realityof q, implies that

while r-invariance leads to

b=, e=h, /=g,

a=b, c=d, e=-h, f=-g.

Hence a, b, c, d are real and e,f, g, h are purely imaginary.Finally we apply SO(2)-invariance. Since i is SO(2)-invariant and is independent

of z,2, ’, ’, we must have q0 and 1 separately SO(2)-invariant. This excludes all termsother than

(A.8) a(z"a+") when a+ 2fl=0,

(A.9) b(z/+’) when a 2/3 0,

(A.10) ie(zO-) when a+2fl=0,

(A.11) if(z’l-) whena-2fl=0,

Now (A.8) and (A.10) imply a fl 0, giving nothing new. The others yield a 2/3.We claim that only a 2, fl 1 yield new generators. For example

(ZX + 2fl+ ..[..,a+2fl+l)=(Zafl.4y,afl)(Z2..,2)__(Z,)2()(Zt-2fl-1 .+. a-2fl- 1).Since b is real and f purely imaginary, we obtain generators

Re(2), ii Im(zZg)in addition to z, N, 8 2. This proves part (a) of Proposition A.1.

Proof of Proposition A.1 (b). Write the general equivariant in the form

where

(z,A)

o" R2 (R2 (R) C) R2,(I) R2 (R2 (R) C) R (R) C

We begin with o- Since the Sl-action on R2 is trivial, the Sl-equivariance conditionimplies that o is Sl-invariant and hence can be written in the form (A.7) above.

However, this time there is no reality condition since we seek mappings into R2,not R. The x-equivariance again implies a,b,c,d are real, and e,f,g,h are purelyimaginary. Replacing the latter by ie, if, ig, ih, we may assume all coefficients a- h arereal, and replace 8 by ii. Write o qo + ibl: again we can treat qo and ql separately.

Now SO(2)-equivariance excludes all terms other than

(A.12) zI, iSz; a+ 2fl= 1,

(A.13) z, iSz"; a- 2fl= 1,

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(A.14) $"’/, i8"’/; a + 2fl= 1,

(A.15) $", i8; -a- 2fl= 1.

In (A.12) we have a= 1, /3=0, yielding z and iSz. In (A.13) we have a= 2fl + 1. Asbefore, we may use the invariance of z$ and ’" to reduce the size of a and fl:

Thus we can reduce/3 by 1 and a by 2. The process stops when/3 1, a 2. But now

Thus we get new generators 5’, ii5’. In (A.14) we can similarly assume fl __< 2. But fl 2gives

+so no new generator arises; and/3= 1 gives 5’ which is already included. Finally (A.15)is not possible.

Thus we have found four generators (z, O)(iSz, 0), (5’, 0), (i85’, 0) corresponding tomappings of 112 (R2 (R) C) into R2.

Now we look at

V R (R (R) C) --, R (R) C.

Again complex notation is more convenient. Define (in a notation consistent with thestatement of Proposition A.1) the complex matrices

(A.16)

H1--(E14-E3)4-i(E2-E4)H2= (E5+E7)+i(E6-E8),n3= ( E E3)4- i(E+ E4)n4= (E ET)-4-i(E6 4- E8).

Then the Sl-equivariants on 112(R) C are generated over C by Hk, /k (k= 1,..., 4) andover R by the real and imaginary parts of Hk (k 1,-.., 4). Since S acts trivially onzR2 we can think of the Sl-equivariants mapping R2(R2(R)C)-R2 as Sl-equivariants mapping R2 (R) C-R2 parametrized by z and 5. Thus they are linear combina-tions of Hk, Hk, (k= 1,.-.,4) with coefficients in C [N, , ’, ’; z,5].

We write the equivariance condition q(3,v)= 3’(v) in the form

(A.17) (v) =’/-x (,v).Suppose

where p" R2 (112 (R) C) R, H: R2 (R2 (R) C)R2 (R) C. Then (A.17) is equivalent to

(A.18) 0(v) H(v) -lp(,v)H()’v) O(3’v)’/-1H(’v).

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We compute this action on Hk (k=l,..-,4) when , O(2). Using (A.16) and notingthat

x(z,A)= ’ 0 -1

(z,A)=(eiq’z,Rq,A)where

we find

(A.19)

sinp)Rq= sin+ cos+

H()

HI HI,H: U//3 H3H4 U

e2in3e2ie/H4

We can write the general Sl-equivariant (I) 1" R2 ( (R (R) C) ---) R2 (8) C in the form

(A.20) {4 )1 =Re E (Pk+Ok)Hkk=l

where the Ok, k are polynomials over C of the form

Ok Pk(N,i, ’, ; z, ), ok=ok(N,8,,; z,5).

Since z2 and ’" are O(2)-invariant, we can write the Ok and ok as

(A.21) az’ + bz’ + cy,’ + dS’t

with a,b,c,d, C[N,(2].We now apply x- and +-equivariance in the form (A.17), writing

o=o(z,A), ,=o(qz,qA).

Now x-equivariance (using (A.18) and (A.19)) implies that

(A.22) )k- k,

(A.23) Ok= --#k

and k-equivariance implies

(A.24) ( 0k (k= 1,2),(A.25) tSk= t -2i4%e Vk (k= 3,4),

(A.26) ( ok (k= 1,2),(A.27) 0k= I -2e iq’o

k (k= 3,4).

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Now write the Pk and % in the form (A.20) (we suppress unnecessary fine points ofnotation in the interests of clarity). From (A.22) we get

(A.28) (for Ok) a=6, b=, c=?, d=a7,

and from (A.23)

(A.29) (for Ok) a=--, b=-, c=-, d=-a.

That is, the coefficients are real for Pk and purely imaginary for %. We thereforereplace ok by iOk, SO that Pk and ok are real: now (A.20) takes the form

(A.30) =Re (Pk +iSk)H,k=l

The +-action multiplies z, ’, and Hk by complex constants e i+, e 2iq’, e 2ik respec-tively. Hence we may consider each of the eight terms in (A.30) separately. From (A.26)and (A.27) we obtain the following conditions on the exponents a, /3, required fork-equivariance:

(A.31)(A.32)(A.33)(A.34)(A.35)(A.36)(A.37)(A.38)

Real part of: k 1, 2 k 3, 4

a+2fl=0a-2fl=0-a+2fl=0-a-2fl=0a+2B=0a-2fl=0-a+2fl=0-a-2fl=0

a+2fl= -2

a-2B -2-a+2fl= -2

-a-2fl= -2a+2fl= -2

a-2fl= -2-a+2= -2

-a-2fl= -2

We deal with these terms case by case, first for k 1, 2; then for k 3, 4. So let k 1, 2.(A.31) implies a =/3 0, leading to the generators

(A.39) Re(Hk), k=l,2.

(A.32) requires a 2fl, so we get

(A.40) z2ttHk +2k.

Similarly (A.33) requires c 2/3, and the result is

(A.41) -aaHk+ z2agak.Forming the sum and difference of (A.40) and (A.41) we may replace them by

+ +

yfl__, ( Z2flfl__ , 2flfl )( Ok k)"We observe the following identities"

x+= 2 Re(Z2)X--(222)X_ 1,

y# + x= 2 Re(z) y#- (zg)y#_ 1,

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which have invariant functions as coefficients. Since

x0=2Re(H,), Yo=0, Xl=2Re(z2)Re(Hg),

and these may be obtained from the generators (A.39), an inductive argument showsthat only Yl need be retained in a list of generators. So we obtain the new generators

(A.42) Im(z2)Im(H,), k 1,2.

For (A.34) we have ct fl 0 and nothing new results.For (A.35) we have et=/3= 0, leading to new generators Re(iSHk), or equivalently

(A.43) 8 Im(H), k 1,2.

From (A.36) and (A.37) we get a 2/3, yielding

Forming the sum and difference, we replace these by

v/= i8( z2O + $2t./)( H,-/k),wt= i8(z2- e2/’/)( H/, +//,).

We note the identities

0,8 + 2 Re(z (z 0,8_1,

Wfl+ 2 Re(z2)w# (z22’)w/_ 1,

Wl=S Xm( z2 )Re( Hk).It follows by induction that no new generators arise here.

Finally (A.38) leads to a=fl=0, and no new generators. This completes theanalysis for k 1, 2.

Next, we let k 3, 4. The calculations follow a similar pattern.

(A.31) is impossible.

(A.32) and (A.33)lead to

u#= 5#+ZaH, + zZ’+ 2#Hk,

Now

tfl + 1--" 2 Re(z2) t/- ( z 252’) tt_ 1’

2 2 Re(z2)t ()U0,U/+ 2 Re(z2) ua- (z 252’) u/_ 1’

u 2 Re(z2) u0- ( z 22) x

(fl>__2),

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Hence inductively the only new generators are ta and u0; that is,

(A.44) ’Hk + ’Hk, k 3,4,

2H+ z2/k, k=3,4.

However, we observe that the identities

NRe( H1) 2 Im(H2 ) Re(H3),26 Im(H1)+NRe(H2) Re(n4)

are valid. Thus the generators .2Hk + z2k (k 3, 4) are redundant and can be omitted.From (A.34) we have either a=0,/3= 1 or a= 2, fl=0. These lead to and u0

again.For convenience we now consider (A.38), for which a 2, /3=0 or c =0, fl= 1.

These lead to new generators

(A.45)i Im(H), k=3,4,

3 Im(2Hk), k=3,4.

Finally we take (A.36) and (A.37), yielding a= 2/3-2 (/3> 1) and c= 2/9 + 2respectively. So we have terms

r13-- ia213-2f13Hk-- i213-213k (fl__> 1),

As usual, we find that

Taking (A.45) into account, we find no new generators.This completes the analysis. We have found twelve generators (A.39), (A.42),

(A.43), (A.44), (A.45). Proposition A.l(b) now follows.Note that the invariants (A.2) for 0(2) S do not form a polynomial ring: there is

a relation

02.._ (Z)2() .._/ 2(N- 432).

Further, the equivariants do not form a free module, although the relations have degree9 or more. For example

[01[ 8 Im(H1) [821 [Im(z 2)Im(/-/) ].

(There are other relations too). In consequence, the singularity theory of O(2)xS onthe six-dimensional kernel would be extremely complicated to compute.

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Finally, we turn to the statement of Theorem 4.3. We obtain the form stated therefor the equivariants from that used in Proposition A.1, by defining

Kj.=Re(Hj), Lj.=Im(Hj.), j=1,2,3,4,

and collecting terms according to the matrices K, L that occur.

Acknowledgments. We are grateful to Bill Langford for suggesting that grouptheory might be useful in the analysis of the six-dimensional kernel. We thank MikeGorman for helping to interpret the experimental evidence and David Andereck forsharing his unpublished observations. John Guckenheimer pointed out some re-dundancies in our original list of generators for the module of equivariant mappings,thus simplifying a number of calculations.

This work was carried out while the second author held a visiting position in theMathematics Department, University of Houston.

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Symmetry and Stability in Taylor-Couette Flow · Introduction. The flow of a fluid between...

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